The Effect of Temperature on the Restitutional Coefficient of a Golf Ball

Subject: IB Physics II Research Project

Researchers: Adam W. Mitchell and Grey M. Patterson

School: Tualatin High School, Tualatin, OR

Supervisor: Mr. Christopher Murray

Submission Date: 13 January 2014

Essay Word Count: 4366

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Table of Contents

I. Introduction .............................................................................................................................................. 1

a. Research Information ........................................................................................................................... 1

b. Background Information ...................................................................................................................... 1

c. Statement of the Problem and Variables ............................................................................................. 2

d. Statement of the Hypothesis ............................................................................................................... 2

II. Method .................................................................................................................................................... 3

a. Experimental Design ............................................................................................................................. 3

b. Materials .............................................................................................................................................. 3

c. Procedure ............................................................................................................................................. 4

d. Illustrations and Diagrams ................................................................................................................... 5

Figure 1: Frontal View of Setup ....................................................................................................... 5

Figure 2: Nitro Golf Balls ................................................................................................................. 6

Figure 3: Top-Down View of Setup ................................................................................................. 6

III. Data ......................................................................................................................................................... 7

a. Collected and Raw Data ....................................................................................................................... 7

Table 1: Initial and Return Heights Part 1 ....................................................................................... 7

Table 2: Initial and Return Heights Part 2 ....................................................................................... 8

b. Calculations and Processed Data ......................................................................................................... 8

Calculation 1: Coefficient of Restitution ......................................................................................... 8

Calculation 2: Average Coefficient of Restitution ........................................................................... 9

Table 3: Coefficients of Restitution and Average Coefficients of Restitution................................. 9

Table 4: Temperatures and Average Coefficients of Restitution .................................................. 10

Graph 1: Temperature vs. Average Coefficients of Restitution (Second Order Trendline) .......... 11

Graph 2: Temperature vs. Average Coefficients of Restitution (Third Order Trendline) ............. 11

Calculation 3: Optimal Temperature (Second Order Trendline) ................................................... 12

Calculation 4: Optimal Temperature (Third Order Trendline) ...................................................... 12

IV. Conclusion ............................................................................................................................................ 13

a. Aspect 1: Overall Conclusion ............................................................................................................. 13

b. Aspect 2: Evaluating Procedures ....................................................................................................... 13

c. Aspect 3: Improving the Investigation ............................................................................................... 15

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V. Works Cited ........................................................................................................................................... 16

VI. Appendices ........................................................................................................................................... 17

a. Appendix A: Original Research Proposal ............................................................................................ 17

b. Appendix B: Preliminary Data Presentation – Notes from Murray .................................................... 18

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I. Introduction

a. Research Information

This research was conducted in Tualatin, Oregon at Tualatin High School, under the supervision

and advice of Mr. Christopher Murray, the IB Physics Instructor. This research project was conceived in

October 2013 by Adam Mitchell and Grey Patterson. The data was collected on Friday, 20 December

2013, and the research paper was completed on Sunday, 5 January 2014. Finally, the research defense

for this investigation was conducted on Monday, 13 January 2013.

b. Background Information

The coefficient of restitution is the measurement of the ratio between the initial drop height of

an object and the height to which the object returns at the apex of its first bounce in its infinite series of

bounces. A perfectly elastic bounce of a ball, in which the ball returns to its original drop height, is

defined as the ball having a coefficient of restitution of 1. A collision in which the ball does not bounce

after the initial bounce, and immediately lies at rest on the floor, is defined as the ball having a

coefficient of restitution of 0 (McGinnis 85). Other definitions and applications of the coefficient of

restitution include the measurement of the ratio between the initial velocity of an object and the final

velocity of an object during its first bounce; however, these irrelevant applications will not be used in

this investigation.

The most common application for the coefficient of restitution, in the modern world, is the area

of sports. Golf balls, soccer, balls, footballs, and all sports balls depend on intricate materials and

precise manufacturing in order to ensure the respective ball behaves in the desired manner. For

example, in the sport of golf, the coefficient of restitution is extremely important, because the sport’s

benchmark conditions specify that a golf ball, in “reasonable temperatures”, is to have a coefficient of

restitution between 0.800 and 0.900 (Thomas). This standard was created in order to attempt to

standardize the frictional, kinetic, and thermal energy loss of the collision between a golf ball and a club.

Therefore, the game of golf greatly values the coefficient of restitution, because the aforementioned

coefficient has a significant impact on the behavior of the respective athletic ball, and therefore, the

outcome of the game.

The one aspect that golf does not account for is whether golf balls can behave differently in

different weather and environmental conditions. Because studying the effects of weather and

environmental conditions on the behavior of golf ball would involve countless variables and

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unimaginable amounts of data, this investigation limited itself to studying the effects of temperature on

the coefficient of restitution of a golf ball. This, however, encompasses a wide range of potential issues.

Firstly, the surface on which the ball bounces has a large impact on the coefficient of restitution

(Horwitz and De). Additionally, because the materials of each golf ball are imperfect, each tested golf

ball will be slightly different in composition and internal pressure (Calsamiglia and Kennedy). More

specifically, Hiroto Kuninaka points out that this investigation must be prepared to account for any

factors that impact the law of conservation of momentum (Kuninaka). Different room air temperatures,

temperature of the bouncing surface, movement of air particles within the experiment room, and

human error will contribute to this experiment’s limitations that this investigation must be prepared to

counter in order to ensure the procurement of valid results.

c. Statement of the Problem and Variables

The purpose of this investigation is to determine the effect that temperature has, if any, on the

coefficient of restitution of a golf ball. Additionally, this investigation will correspondingly answer the

question, “at what optimal temperature should a golf ball be dropped in order to produce the highest

coefficient of restitution?” In other words, this investigation will attempt to conclude at which

temperature a golf ball loses the least amount of energy to an elastic collision.

d. Statement of the Hypothesis

This investigation believes that the graphical results of this experiment will resemble a bell

curve, with the lower temperatures producing relatively low coefficients of restitution, the higher

temperatures producing relatively low coefficients of restitution, and the reasonable and moderate

temperatures producing relatively higher coefficients of restitutions, because the athletic golf balls were

designed for optimal performance in moderate temperatures. This hypothesis is a result of the

assumption that the designers of the golf balls attempted to optimize the balls for minimal energy loss

at typical temperatures due to the moderate thermal conditions of a game of golf. This investigation

furthermore relies on the assumption that the designers of the golf balls did not attempt to reduce the

amount of energy lost during an elastic collision at extreme temperatures, because a game of golf is

typically not played in extreme thermal conditions.

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II. Method

a. Experimental Design

This investigation will achieve its aims of determining the effect of temperature on the

coefficient of restitution of a golf ball by examining the drop height, return height, and initial

temperature of a golf ball over multiple trials. Different golf balls will be used for different temperatures

in order to ensure precise results. Additionally, the computerized system LoggerPro will analyze and

interpret the drop height and return height in order to eliminate the significant factor of human

measurement error.

b. Materials

The following is a list of materials that were used by this investigation.

24 golf balls1

5-pound block of dry ice (to cool golf balls)

2 1000 ml beakers (to hold water and golf balls)

2 hot plates (to heat water)

2 temperature probes (to record temperature)

1 computer with LoggerPro (to record readings of temperature probes)

1 faucet / sink (to procure water)

1 hammer (to break dry ice block)

2 gloves (to handle dry ice block)

1 pair of forceps (to handle dry ice and golf balls)

1 insulated cooler (to preserve dry ice block)

1 camera with video capabilities (to record videos to be later analyzed)

2 meter sticks (to measure drop and return heights)

1 The golf balls were manufactured by the company “Nitro.” A picture can be found in subsection “d. Illustrations and Diagrams” of section “II. Method.”

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c. Procedure

In order to properly perform this experiment, one must first gather the aforementioned

materials. Once this is completed, the experimenter must take steps to properly ensure his/her physical

safety and to arrange the experimental setup. This is accomplished by placing gloves on the

experimenter’s hands, placing the dry ice block in the insulated cooler and using the hammer to break

the ice into reasonably small pieces, filling each 1000 ml beaker to 600 ml of water, placing each beaker

on its own hot plate, connecting each hotplate to an electrical outlet, turning on each hotplate, placing 1

temperature probe in each of the beakers on the hotplates, connecting the temperature probes to the

computer, and opening LoggerPro in order to monitor the temperature of the two beakers. Note that

these hotplates and beakers will later be used to change the temperatures of the golf balls to the

desired values. Next, manipulate the heat of the first hot plate in order to bring the temperature of the

water in its beaker to 100 ± 1 ℃. At the same time, manipulate the heat of the second hot plate to 73.5

± 1 ℃. Once each temperature has stabilized at the aforementioned values, one should place three golf

balls into each beaker.2 Then, the experimenter should place three golf balls in the dry ice cooler and

ensure that each golf ball is in direct, physical contact with the dry ice. Once the various golf balls begin

to acclimate to their respective medium’s temperature, the experimenter should tape 1 meter stick to a

counter so that the meter stick is in an upright, vertical position in order to measure a ball’s height when

dropped.3 After this, he/she should place a piece of tape on top of the counter, 15 cm from and parallel

to the edge of the counter, and mark that location for it will be from where the golf balls will be pushed

off of the counter. For the final preparation of the setup, place the camera on a stable surface so that

the camera’s frame and view can see the entirety of the experiment’s frontal setup. The setup process

should now be complete, and if more assistance is required, please consult subsection “d. Illustrations

and Diagrams” of section “II. Method.”

To begin collecting data, select three golf balls that have been at room temperature (20.5 ± 1

℃) and set them aside. Of these three golf balls, select one and set it behind the starting line, 15 cm

back from the edge of the counter. In order to accurately record the experiment and data collection,

the experimenter should commence the camera’s video recording by pressing the appropriate button.

Once the camera is recording, an experimenter should gently push the golf ball off of the counter,

towards the camera. The experimenters should wait for the ball to bounce twice, and then stop the

2 ALL GOLF BALLS SHOULD STAY IN THEIR RESPECTIVE MEDIUM WITH THE CORRECT TEMPERATURE FOR APPROXIMATELY TEN MINUTES. 3 See subsection “d. Illustrations and Diagrams” of section “II. Method” for a picture and diagram.

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camera’s video recording by pressing the appropriate button. Using the same golf ball, the

experimenters should repeat the process of recording data with the appropriate procedures for an

additional two trials, thereby producing a total of three trials per each golf ball. Once this is complete,

the experimenters should repeat the data collection steps with each of the other two golf balls that

were in the original group of three golf balls at the same temperature and in the same medium. This

should produce a grand total of 9 measurements (3 trials each of the 3 golf balls) for each temperature.

The experimenters should then repeat this process of data collection for each of the golf balls in the

remaining temperatures: the dry ice golf balls, the 100 ± 1 ℃ golf balls, and the 73.5 ± 1 ℃ golf balls.

Once this data collection is complete, the beakers and setup will need to be adjusted in order to

produce more testable temperatures for this experiment. To begin, the experimenter must

mannipulate the temperature of the first beaker (100 ± 1 ℃) in order to achieve a temperature of 61.0

± 1 ℃. This may involve sublimating a small amount of dry ice in the beaker in order to cool the water.

Simultaneously, he/she must manipulate the temperature of the second beaker (73.5 ± 1 ℃) in order to

achieve a temperature of 17.5 ± 1 ℃. This may also involve sublimating a small amount of dry ice in the

beaker in order to cool the water. Once each of the water temperatures stabilize at the aforementioned

values, place three golf balls into each beaker for this experiment’s temporal standard of approximately

ten minutes. Once the golf balls have acclimated to their respective mediums’ temperatures, the

experimenters should repeat the data collection steps with the 61.0 ± 1 ℃ golf balls, and then the 17.5

± 1 ℃ golf balls. After this, the experimenters should clean-up the setup and put away all materials.

Finally, This procedure culminates with the experimenter appropriately analyzing each video with

LoggerPro in order to determine each ball’s drop and return heights.

d. Illustrations and Diagrams

Figure 2: Frontal View of Setup

Figure 1: Frontal View of Setup

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Vertical Meter Stick

Piece of Tape

Counter

15 cm

Figure 3: Top-Down View of Setup

Golf Ball

Figure 2: Nitro Golf Balls

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III. Data

a. Collected and Raw Data4

Table 1: Initial and Return Heights Part 15

4 Due to the position of the camera, which was about one-half meter off of the ground during filming, the LoggerPro program incorrectly triangulated the location of the golf ball due to the program’s assumption that the camera was resting on the ground during filming. Therefore, LoggerPro, and this investigation’s data, states that the initial drop height of most golf balls was near 1.5 meters, although “Figure 1: Frontal View of Setup” in subsection “d. Illustrations and Diagrams” of section “II. Method” clearly shows the ball’s initial height to be less than a meter. However, the LoggerPro program also incorrectly triangulated the return height of the ball. Therefore, the overall coefficient of restitution for a given ball was unaffected because the coefficient of restitution is simply calculated by dividing the return height by the drop height. If both of these values are translated in one direction by the same magnitude, which they were by incorrect triangulation, then the coefficient of restitution remained unaffected. 5 The recordings of Dry Ice Ball 2 (trials 1 – 3) were lost by accidental deletion.

Ball Type / Temperature (℃) Ball Trial Initial Height (m) ± .05 m Final Height (m) ± .05 m

Room Temp (20.5 ± 1 ℃) 1 1 1.520 1.309

Room Temp (20.5 ± 1 ℃) 1 2 1.536 1.294

Room Temp (20.5 ± 1 ℃) 1 3 1.545 1.317

Room Temp (20.5 ± 1 ℃) 2 1 1.555 1.338

Room Temp (20.5 ± 1 ℃) 2 2 1.550 1.328

Room Temp (20.5 ± 1 ℃) 2 3 1.532 1.293

Room Temp (20.5 ± 1 ℃) 3 1 1.577 1.352

Room Temp (20.5 ± 1 ℃) 3 2 1.554 1.311

Room Temp (20.5 ± 1 ℃) 3 3 1.527 1.299

Dry Ice (-78.5 ± 1 ℃) 1 1 1.516 1.118

Dry Ice (-78.5 ± 1 ℃) 1 2 1.512 1.152

Dry Ice (-78.5 ± 1 ℃) 1 3 1.548 1.183

Dry Ice (-78.5 ± 1 ℃) 3 1 1.545 1.194

Dry Ice (-78.5 ± 1 ℃) 3 2 1.558 1.213

Dry Ice (-78.5 ± 1 ℃) 3 3 1.545 1.172

Boiling (100 ± 1 ℃) 1 1 1.582 1.184

Boiling (100 ± 1 ℃) 1 2 1.553 1.169

Boiling (100 ± 1 ℃) 1 3 1.598 1.240

Boiling (100 ± 1 ℃) 2 1 1.600 1.217

Boiling (100 ± 1 ℃) 2 2 1.577 1.197

Boiling (100 ± 1 ℃) 2 3 1.544 1.227

Boiling (100 ± 1 ℃) 3 1 1.593 1.198

Boiling (100 ± 1 ℃) 3 2 1.591 1.181

Boiling (100 ± 1 ℃) 3 3 1.561 1.160

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Table 2: Initial and Return Heights Part 2

b. Calculations and Processed Data

Calculation 1: Coefficient of Restitution

𝐶𝑜𝑒𝑓𝑅𝑒𝑠𝑡 =𝐻𝑒𝑖𝑔ℎ𝑡𝐷

𝐻𝑒𝑖𝑔ℎ𝑡𝑅

Where 𝐶𝑜𝑒𝑓𝑅𝑒𝑠𝑡 is the Coefficient of Restitution (constant / scalar / no units)

𝐻𝑒𝑖𝑔ℎ𝑡𝐷 is the initial drop height of the golf ball in meters (m)

Ball Type / Temperature (℃) Ball Trial Initial Height (m) ± .05 m Final Height (m) ± .05 m

Hot (73.5 ± 1 ℃) 1 1 1.572 1.355

Hot (73.5 ± 1 ℃) 1 2 1.569 1.361

Hot (73.5 ± 1 ℃) 1 3 1.565 1.367

Hot (73.5 ± 1 ℃) 2 1 1.569 1.362

Hot (73.5 ± 1 ℃) 2 2 1.566 1.362

Hot (73.5 ± 1 ℃) 2 3 1.577 1.376

Hot (73.5 ± 1 ℃) 3 1 1.580 1.345

Hot (73.5 ± 1 ℃) 3 2 1.544 1.340

Hot (73.5 ± 1 ℃) 3 3 1.565 1.362

Mild (61.0 ± 1 ℃) 1 1 1.542 1.339

Mild (61.0 ± 1 ℃) 1 2 1.562 1.365

Mild (61.0 ± 1 ℃) 1 3 1.569 1.371

Mild (61.0 ± 1 ℃) 2 1 1.574 1.371

Mild (61.0 ± 1 ℃) 2 2 1.576 1.355

Mild (61.0 ± 1 ℃) 2 3 1.520 1.286

Mild (61.0 ± 1 ℃) 3 1 1.573 1.376

Mild (61.0 ± 1 ℃) 3 2 1.592 1.361

Mild (61.0 ± 1 ℃) 3 3 1.546 1.356

Cold (17.5 ± 1 ℃) 1 1 1.572 1.325

Cold (17.5 ± 1 ℃) 1 2 1.589 1.374

Cold (17.5 ± 1 ℃) 1 3 1.559 1.339

Cold (17.5 ± 1 ℃) 2 1 1.590 1.355

Cold (17.5 ± 1 ℃) 2 2 1.583 1.383

Cold (17.5 ± 1 ℃) 2 3 1.563 1.344

Cold (17.5 ± 1 ℃) 3 1 1.565 1.333

Cold (17.5 ± 1 ℃) 3 2 1.588 1.362

Cold (17.5 ± 1 ℃) 3 3 1.586 1.376

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𝐻𝑒𝑖𝑔ℎ𝑡𝑅 is the return height of the gold ball in meters (m)

Calculation 2: Average Coefficient of Restitution

𝐴𝑣𝑟𝑔𝐶𝑜𝑒𝑓𝑅𝑒𝑠𝑡= 𝑛−1 ∙ ∑ 𝐶𝑜𝑒𝑓𝑅𝑒𝑠𝑡

𝑛

1

Where 𝐴𝑣𝑟𝑔𝐶𝑜𝑒𝑓𝑅𝑒𝑠𝑡 is the Coefficient of Restitution (constant / scalar / no units)

𝑛 is the total number of golf ball drops for a specific temperature6

𝐶𝑜𝑒𝑓𝑅𝑒𝑠𝑡 is the Coefficient of Restitution (constant / scalar / no units)

Table 3: Coefficients of Restitution and Average Coefficients of Restitution

Ball Type / Temperature (℃) Ball Trial Coef. Of Rest. Avrg. Coef. Of Rest.

Room Temp (20.5 ± 1 ℃) 1 1 0.861184211 0.852102

Room Temp (20.5 ± 1 ℃) 1 2 0.842447917

Room Temp (20.5 ± 1 ℃) 1 3 0.852427184

Room Temp (20.5 ± 1 ℃) 2 1 0.860450161

Room Temp (20.5 ± 1 ℃) 2 2 0.856774194

Room Temp (20.5 ± 1 ℃) 2 3 0.843994778

Room Temp (20.5 ± 1 ℃) 3 1 0.857324033

Room Temp (20.5 ± 1 ℃) 3 2 0.843629344

Room Temp (20.5 ± 1 ℃) 3 3 0.850687623

Dry Ice (-78.5 ± 1 ℃) 1 1 0.737467018 0.762256

Dry Ice (-78.5 ± 1 ℃) 1 2 0.761904762

Dry Ice (-78.5 ± 1 ℃) 1 3 0.764211886

Dry Ice (-78.5 ± 1 ℃) 3 1 0.772815534

Dry Ice (-78.5 ± 1 ℃) 3 2 0.778562259

Dry Ice (-78.5 ± 1 ℃) 3 3 0.758576052

Boiling (100 ± 1 ℃) 1 1 0.748419722 0.758770

Boiling (100 ± 1 ℃) 1 2 0.752736639

Boiling (100 ± 1 ℃) 1 3 0.775969962

Boiling (100 ± 1 ℃) 2 1 0.760625000

Boiling (100 ± 1 ℃) 2 2 0.759036145

Boiling (100 ± 1 ℃) 2 3 0.794689119

Boiling (100 ± 1 ℃) 3 1 0.752040176

Boiling (100 ± 1 ℃) 3 2 0.742300440

6 In all cases but the Dry Ice Temperature, where n is 6, n is 9.

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Boiling (100 ± 1 ℃) 3 3 0.743113389

Hot (73.5 ˚C) 1 1 0.861959288 0.866961

Hot (73.5 ˚C) 1 2 0.867431485

Hot (73.5 ˚C) 1 3 0.873482428

Hot (73.5 ˚C) 2 1 0.868068834

Hot (73.5 ˚C) 2 2 0.869731801

Hot (73.5 ˚C) 2 3 0.872542803

Hot (73.5 ˚C) 3 1 0.851265823

Hot (73.5 ˚C) 3 2 0.867875648

Hot (73.5 ˚C) 3 3 0.870287540

Mild (61.0 ˚C) 1 1 0.868352789 0.866628

Mild (61.0 ˚C) 1 2 0.873879641

Mild (61.0 ˚C) 1 3 0.873804971

Mild (61.0 ˚C) 2 1 0.871029225

Mild (61.0 ˚C) 2 2 0.859771574

Mild (61.0 ˚C) 2 3 0.846052632

Mild (61.0 ˚C) 3 1 0.874761602

Mild (61.0 ˚C) 3 2 0.854899497

Mild (61.0 ˚C) 3 3 0.877102199

Cold (17.5 ˚C) 1 1 0.842875318 0.858803

Cold (17.5 ˚C) 1 2 0.864694777

Cold (17.5 ˚C) 1 3 0.858883900

Cold (17.5 ˚C) 2 1 0.852201258

Cold (17.5 ˚C) 2 2 0.873657612

Cold (17.5 ˚C) 2 3 0.859884837

Cold (17.5 ˚C) 3 1 0.851757188

Cold (17.5 ˚C) 3 2 0.857682620

Cold (17.5 ˚C) 3 3 0.867591425

Table 4: Temperatures and Average Coefficients of Restitution

Ball Type / Temperature (℃) Avrg. Coef. Of Rest.

Dry Ice (-78.5 ± 1 ℃) 0.762256

Cold (17.5 ˚C) 0.858803

Room Temp (20.5 ± 1 ℃) 0.852102

Mild (61.0 ˚C) 0.866628

Hot (73.5 ˚C) 0.866961

Boiling (100 ± 1 ℃) 0.758770

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Graph 1: Temperature vs Average Coefficients of Restitution (Second Order Trendline)7

Graph 2: Temperature vs Average Coefficients of Restitution (Third Order Trendline)

7 Errors and uncertainties in both the x-direction (Temperature) and y-direction (Avrg. Coef. Of Rest.) were negligible at this scale.

y = -1E-05x2 + 0.0004x + 0.8674

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

-100 -80 -60 -40 -20 0 20 40 60 80 100 120

Avr

g. C

oe

f. o

f R

est

.

Temperature in Degrees Celsius

y = -2E-07x3 - 3E-06x2 + 0.0019x + 0.8218

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

-100 -80 -60 -40 -20 0 20 40 60 80 100 120

Avr

g. C

oe

f. o

f R

est

.

Temperature in Degrees Celsius

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Calculation 3: Optimal Temperature (Second Order Trendline)

The optimal temperature for the highest coefficient of restitution, using the second order

trendline, can be found be taking the derivative of the second order trendline function, setting it equal

to 0, and solving for x.

𝑓(𝑥) = −0.00001𝑥2 + 0.0004𝑥 + .8674

𝑓′(𝑥) = −0.00002𝑥 + 0.0004

0 = −0.00002𝑥 + 0.0004

0.00002𝑥 = 0.0004

0.00002𝑥 = 0.0004

𝑥 = 20.0

Therefore, the optimal temperature for the highest coefficient of restitution, using the second

order trendling, is 20.0 ℃.

Calculation 4: Optimal Temperature (Third Order Trendline)

The optimal temperature for the highest coefficient of restitution, using the third order

trendline, can be found be taking the derivative of the third order trendline function, setting it equal to

0, and solving for x.

𝑓(𝑥) = −0.0000002𝑥3 − 0.000003𝑥2 + 0.0019𝑥 + 0.8218

𝑓′(𝑥) = −0.0000006𝑥2 − 0.000006𝑥 + 0.0019

0 = −0.0000006𝑥2 − 0.000006𝑥 + 0.0019

𝑥 = 51.4948, −61.4948

Therefore, the optimal temperature for the highest coefficient of restitution, using the second

order trendline, is 51.4948℃. Note that the solution -61.4948 is discarded, because the graph supports

the conclusion that this temperature produces the lowest coefficient of restitution possible.

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IV. Conclusion

a. Aspect 1: Overall Conclusion

This investigation’s hypothesis was ultimately supported in that the temperature vs. coefficient

of restitution graph and data points resembled a bell curve which peaked between 20 – 60 ℃. Strictly

according to the graph, the 73.5 ± 1 ℃ produced the highest coefficient of restitution. However, the

difference between the coefficients of restitution produced by the 73.5 ± 1 ℃ temperature and the

room temperature (20.5 ± 1 ℃) was approximately 0.014859. Converting that value to meters by

multiplying by the initial drop height, the difference between the heights produced by the 73.5 ± 1 ℃

temperature and the room temperature (20.5 ± 1 ℃) was approximately 2.22885 cm. Therefore, this

investigation concludes that temperatures within the reasonable and approximate range of 0 – 75 ℃,

produce extremely similar coefficients of restitution.

Additionally, according to the second order trendline, the optimal temperature for the highest

coefficient of restitution was 20 ℃, or 68 ℉. According to the third order trendline, the optimal

temperature for the highest coefficient of restitution was 51.4948 ℃. Thus, both of these trendlines

support the hypothesis that “reasonable temperatures” (Thomas) produce the highest coefficients of

restitution when compared to coefficients of restitution produced by extreme thermal values. However,

because the data points do not clearly distinguish the best trendline (second order vs. third order), this

investigation ventures to logically conclude that more data points would be extremely helpful in

determining the appropriate graph to resemble the data points. Nevertheless, in order to determine

the overall optimum temperature for the highest coefficient of restitution, this investigation averaged

the optimum values provided by the second order and third order trendlines. This overall optimum

temperature estimate was found to be ((20 + 51.4948) / 2) 35.7474 ℃, or 96.3453 ℉. Therefore, while

there are noteworthy limitations to this experiment, this investigation concludes that its hypothesis was

supported by the shape of the graph, the results of the data, and the approximations of the trendlines.

b. Aspect 2: Evaluating Procedures

There were many procedural flaws and areas for improvement in this investigation. The most

prominent procedural weakness was that the experiment relied upon the assumption that the golf balls

were the same temperature as that of medium in which they were placed. For example, this inquiry

assumed that the temperature of the dry ice golf balls was -78.5 ± 1 ℃. However, this temperature

recording was that of the dry ice itself. Therefore, while the phenomenon of thermal contact and the

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field of thermodynamics support the assumption that the golf balls were relatively close to the

temperature of their medium, this investigation must nevertheless recognize this imperfection in the

procedure.

The second potential weakness was that each golf ball spent a different amount of time in its

respective medium. For example, this investigation’s procedure mandated that all golf balls be placed in

thermal contact with their medium, once the medium’s temperature had stabilized at the appropriate

value, for 10 minutes. After 10 minutes, the procedure instructed one golf ball to be removed from the

medium in order to conduct 3 trials with the golf ball. Therefore, the other two golf balls in the medium

were in thermal contact with their medium for a longer duration of time than the first. It follows that

the third golf ball was in thermal contact with its medium for the longest period of time, while the first

golf ball was in thermal contact with the same medium for the shortest period of time. According to

Newton’s Law of Cooling, heat transfer occurs extremely quickly after immediate thermal contact and

gradually slows after thermal contact has been experienced for some time. Thus, while this

investigation discounts the possibility of this procedural error altering the entirety of the results and

conclusion, it must be noted that placing an experiment’s objects in a respective medium for varying

amounts of time is a potential procedural flaw.

The most practical weakness in this investigation’s procedure is that three trials were conducted

with each golf ball. Accordingly, the golf ball contained more heat, and therefore energy, during the first

trial than during the third trial. This is a significant possible error due to Newton’s Law of Cooling. Apart

from the aforementioned aspects of Newton’s Law of Cooling, this law also states that the rate of

change of heat transfer is dependent upon the difference between the environment and the object. For

example, if a relatively hot object (15 – 20 ℃) was placed in a relatively cold environment (0 – 5 ℃),

then the object would lose heat at a relatively slow rate, because the difference between the object and

the environment is relatively small. Therefore, when the extremely cold dry ice golf balls (-78.5 ± 1 ℃),

in this experiment, were placed in a room temperature environment (20.5 ± 1 ℃), the balls would gain

heat relatively quickly. Additionally, when the extremely hot golf balls (100.5 ± 1 ℃), were placed in a

room temperature environment (20.5 ± 1 ℃), the balls would lose heat relatively quickly. Thus, the

danger that this procedure incurred was that three trials with each golf ball provided each golf ball with

the opportunity to have an extremely different heat content during the third trial than during the first.

Because the three trials of each golf ball took less than 2 minutes to complete, the thermal difference

between the first and third trial is predicted to be relatively low. Nevertheless, this investigation must

recognize this potential procedural flaw.

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c. Aspect 3: Improving the Investigation

There are several measures that could be taken in order to improve the aspects of this

investigation. Firstly, placing 9 golf balls in each thermal medium would provide the same amount of

data for each medium when compared to placing 3 golf balls in each medium and performing 3 trials

with each golf ball. The advantage to using 9 golf balls in each medium, instead of 3, is that only one

trial would be performed on each golf ball. This would eliminate the potential thermal difference

between a golf ball’s first and third trials as only one trial per golf ball would occur. This, in turn, would

provide more accurate and reliable results.

Secondly, this investigation would be improved if the experiment was conducted in the

conditions of Standard Temperature and Pressure8. For practicality, this investigation was conducted in

an environment with a temperature of 20.5 ± 1 ℃ (68.9 ± 1.8 ℉) and a pressure of 1.00 atm (in

absolute pressure). Because these conditions significantly differ from the conditions of Standard

Temperature and Pressure, this investigation would produce more reliable and standardized results if

the experiment was conducted in the conditions of Standard Temperature and Pressure.

Furthermore, another potential improvement to this investigation is for the golf balls to be

placed in their respective mediums for a longer duration of time than 10 minutes. While this would

ensure that the golf balls are relatively close to the temperature of the medium, Newton’s Law of

Cooling assured this investigation that 10 minutes would suffice for the temperature differences that

this investigation used. Note that Newton’s Law of Cooling, it its differential form, is 𝑑𝑦

𝑑𝑥=

−𝑘 ∙ 𝑦 .

Finally, the largest improvement that could be made to this investigation is to use an insulated

thermal contact probe in order to quickly measure the temperature of the golf ball. This would reduce

the discrepancy between measuring the medium of the golf ball and assuming that the golf balls share

the temperature of their medium. Ultimately, this investigation concludes that these potential

procedural flaws did not severely affect the results of this experiment. However, the improvement of

this investigation and its procedure could yield more effective results.

8 The Standard Conditions for Temperature and Pressure are an international set of standards that was created by the International Union of Pure and Applied Chemistry (IUPAC). The conditions specify the temperature and pressure that an environment must be in order to conduct standard and reliable experiments. The conditions are as follows. Temperature: 0 ℃, 32 ℉, 273.15 K. Pressure: 0.987 atm, 14.504 psi, , 1 bar (in absolute pressure).

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V. Works Cited

Calsamiglia, J and S. W. Kennedy. ""Anomalous Frictional Behavior in Collisions of Thin Disks." Journal of

Applied Mechanics (2002): 66.

Horwitz, Dr. James L. and Dr. Kaushik De. "For Immediate Release - October 28, 2006." 28 October 2006.

American Physical Society. 22 December 2013

<http://www.aps.org/about/pressreleases/20061028.cfm>.

Kuninaka, Hiroto. "Anomalous Behavior of the Coefficient of Normal Restitution in Oblique Impact."

Physical Review Letters (2003): 93.

McGinnis, Peter Merto. Biomechanics of Sport and Exercise. Champaign, Illinois: Human Kinetics, 2005.

Thomas, Frank. "Everything You Need to Know About COR." Golf Digest (2002).

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VI. Appendices

a. Appendix A: Original Research Proposal

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b. Appendix B: Preliminary Data Presentation – Notes from Murray